How to place circles that there will be no gap. Use least number of circles possible.by Firewolffzc Tags: circle 

#1
Apr2112, 02:34 PM

P: 7

I have to plan the layout of a sprinkler system. Basically, each sprinkler shoots a radius of 7.5 feet water, and I want every part of the floor covered with water. How can I use the least number of sprinklers?




#2
Apr2112, 02:52 PM

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Hi Firewolffzc! Welcome to PF!
Tell us what you think, and then we'll comment! 



#3
Apr2112, 02:54 PM

P: 7

Well, I think I figured one way to approach. Is it better to imagine not the circle, but a triangle, square or hexagon inscribed within the circle, because these shapes can link without gaps. Then once done with these shapes, I draw a circle circumscribing the shape. I was wondering if this was an effective method. If it is, which shape is best?




#4
Apr2112, 03:02 PM

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How to place circles that there will be no gap. Use least number of circles possible.
yup, that seems a good idea!
i'd guess it's the hexagon, but you'd better do the maths! 



#5
Apr2212, 11:52 AM

P: 7

Well, see, I don't know what math to do! I figured out that the reason we can use triangles, squares and hexagons is because the measurement of their angles are multiples of 360. (triangle)60*6=360, (square)90*4=360, (hexagon)120*3= 360. Well, it is not possible to have a shape with a 180 or 360 degrees angle, therefore hexagons are the best shape because they make the smallest area between circles. Wow, I can't believe I figured that out myself. Anyway, is there more to it or is this the final solution?




#6
Apr2212, 01:55 PM

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no, that seems fine, so long as you actually prove …




#7
Apr2212, 05:20 PM

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A hexagonal packing might be the best for covering the entire plane (not sure), but for finite size rooms, an irregular packing can do better
This is a page with the best known results for covering a square with circles http://www2.stetson.edu/~efriedma/circovsqu/ 



#9
Apr2312, 06:38 PM

P: 7

Thanks for all your help. Btw, I don't know how to prove something like the area formed between circles circumscribing hexagons is the smallest compared to triangles and squares.



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